A Local Version of Katona's Intersecting Shadow Theorem

Katona's intersection theorem states that every intersecting family F ⊆ [n]ᵏ satisfies |∂F| ⩾ |F|, where ∂F = {F\ {x} : x ϵ F ϵ F} is the shadow of F. Frankl conjectured that for n > 2k and every intersecting family F ⊆ [n]^⁽ᵏ⁾, there is some i ∈ [n] such that |∂F(i)| ⩾ |F(i)|, where F(i)={F∖ {i} : i ∈ F ∈ F} is the link of F at i. Here, we prove this conjecture in a very strong form for n > (^(k+1)_(2)). In particular, our result implies that for any j ∈ [k], there is a j-set {a₁, . . . , a_j} ∈ [n]⁽ʲ⁾ such that |∂F(a₁, . . . , a_j)| ⩾ |F(a₁, . . . , a_j)|. A similar statement is also obtained for cross-intersecting families.